fundamental-engineering-optimization-methods.pdf

Due to the fact that convex functions have a unique

Info icon This preview shows pages 25–27. Sign up to view the full content.

Due to the fact that convex functions have a unique global minimum, convexity plays an important role in optimization. For example, in numerical optimization convexity assures a global minimum to the problem. It is therefore important to first establish the convexity property when solving optimization problems. The following characterization of convexity applies to the solution spaces in such problems. Further ways of establishing convexity are discussed in (Boyd & Vandenberghe, Chaps. 2&3). If a function ݃ ሺ࢞ሻ is convex, then the set ݃ ሺ࢞ሻ ൑ ݁ is convex. Further, if functions ݃ ሺ࢞ሻǡ ݅ ൌ ͳǡ ǥ ǡ ݉ǡ are convex, then the set ሼ࢞ǣ ݃ ሺ࢞ሻ ൑ ݁ ǡ ݅ ൌ ͳǡ ǥ ǡ ݉ሽ is convex. In general, finite intersection of convex sets (that include hyperplanes and halfspaces) is convex. For general optimization problems involving inequality constraints: ݃ ሺ࢞ሻ ൑ ݁ ǡ ݅ ൌ ͳǡ ǥ ǡ ݉ ³ DQG ݈ , and equality constraints: ݄ ሺ࢞ሻ ൌ ܾ ǡ ݆ ൌ ͳǡ ǥ ǡ ݈ǡ W the feasible region for the problem is defined by the set: ܵ ൌ ሼ࢞ǣ ݃ ሺ࢞ሻ ൑ ݁ ǡ ݄ ሺ࢞ሻ ൌ ܾ ² The feasible region is a convex set if the functions: ݃ ݅ ൌ ͳǡ ǥ ǡ ݉ǡ are convex and the functions: ݄ ǡ ݆ ൌ ͳǡ ǥ ǡ ݈ǡ are linear. Note that these convexity conditions are sufficient but not necessary. Visit us and find out why we are the best! Master’s Open Day: 22 February 2014 Join the best at the Maastricht University School of Business and Economics! Top master’s programmes • 33 rd place Financial Times worldwide ranking: MSc International Business • 1 st place: MSc International Business • 1 st place: MSc Financial Economics • 2 nd place: MSc Management of Learning • 2 nd place: MSc Economics • 2 nd place: MSc Econometrics and Operations Research • 2 nd place: MSc Global Supply Chain Management and Change Sources: Keuzegids Master ranking 2013; Elsevier ‘Beste Studies’ ranking 2012; Financial Times Global Masters in Management ranking 2012 Maastricht University is the best specialist university in the Netherlands (Elsevier)
Image of page 25

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

Download free eBooks at bookboon.com Fundamental Engineering Optimization Methods 26 Mathematical Preliminaries 2.6 Vector and Matrix Norms Norms provide a measure for the size of a vector or matrix, similar to the notion of absolute value in the case of real numbers. A norm of a vector or matrix is a real-valued function with the following properties: 1. ԡ࢞ԡ ൒ Ͳ for all כ 2. ԡ࢞ԡ ൌ Ͳ if and only if ࢞ ൌ ૙ 3. ԡߙ࢞ԡ ൌ ȁߙȁԡ࢞ԡ for all ߙ א Թ 4. ԡ࢞ ൅ ࢟ԡ ൑ ԡ࢞ԡ ൅ ԡ࢟ԡ Matrix norms additionally satisfy: 5. || AB ||≤|| A || || B || Vector Norms. Vector p-norms are defined by ԡ࢞ԡ ൌ ሺσ ȁݔ ȁ ௜ୀଵ ǡ ݌ ൒ ͳ ² They include the 1-norm ԡ࢞ԡ ൌ σ ȁݔ ȁ ௜ୀଵ ³ the Euclidean norm P ԡ࢞ԡ ൌ ඥσ ȁݔ ȁ ௜ୀଵ ³
Image of page 26
Image of page 27
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern